Module: FasterPrime::Sieve

Defined in:
lib/faster_prime/sieve.rb

Constant Summary collapse

TBL = 
[]
MODULO60 = 
[1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59]

Class Method Summary collapse

Class Method Details

.each(ubound = nil, &blk) ⇒ Object 



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# File 'lib/faster_prime/sieve.rb', line 26

def each(ubound = nil, &blk)
  if ubound && SMALL_PRIMES.last >= ubound
    SMALL_PRIMES.each do |n|
      break if n > ubound
      yield n
    end
    return
  end

  u = SMALL_PRIMES.last / 60 * 60
  SMALL_PRIMES.pop until SMALL_PRIMES.last < u
  SMALL_PRIMES.each(&blk)

  b = Utils.integer_square_root(ubound * 3000) if ubound
  b = 100000 if !b || b > 100000
  l = u / 60

  # Algorithm 3.1, 3.2, 3.3
  flags = [0] * b
  while true
    # step
    flags.fill(0)

    # sieve
    TBL.each do |name, f, g, d|
      send(name, flags, f, g, d, l, b)
    end

    # squarefree
    squarefree(flags, l, b)

    # enumerate primes
    if ubound && ubound < 60 * (l + b)
      enumerate(flags, l, b) do |p|
        return if p >= ubound
        yield p
      end
    else
      enumerate(flags, l, b, &blk)
    end

    l += b
  end
end

.enumerate(flags, l, b) ⇒ Object 



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# File 'lib/faster_prime/sieve.rb', line 71

def enumerate(flags, l, b)
  b.times do |x|
    base = 60 * (l + x)
    flag = flags[x]
    MODULO60.each do |d|
      next if flag[d] == 0
      p = base + d
      SMALL_PRIMES << p
      yield p
    end
  end
end

.mark_type_1(flags, f, g, d, l, b) ⇒ Object

Algorithm 4.1



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# File 'lib/faster_prime/sieve.rb', line 104

def mark_type_1(flags, f, g, d, l, b)
  x, y0 = f, g
  k0 = (4 * f * f + g * g - d) / 60 - l
  m = 1 << d
  while k0 < l + b
    k0 += 2*x + 15
    x += 15
  end
  while true
    x -= 15
    k0 -= 2*x + 15
    return if x <= 0
    while k0 < 0
      k0 += y0 + 15
      y0 += 30
    end
    k, y = k0, y0
    while k < b
      flags[k] ^= m # mark
      k += y + 15
      y += 30
    end
  end
end

.mark_type_2(flags, f, g, d, l, b) ⇒ Object

Algorithm 4.2



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# File 'lib/faster_prime/sieve.rb', line 130

def mark_type_2(flags, f, g, d, l, b)
  x, y0 = f, g
  k0 = (3 * f * f + g * g - d) / 60 - l
  m = 1 << d
  while k0 < b
    k0 += x + 5
    x += 10
  end
  while true
    x -= 10
    k0 -= x + 5
    return if x <= 0
    while k0 < 0
      k0 += y0 + 15
      y0 += 30
    end
    k, y = k0, y0
    while k < b
      flags[k] ^= m # mark
      k += y + 15
      y += 30
    end
  end
end

.mark_type_3(flags, f, g, d, l, b) ⇒ Object

Algorithm 4.3



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# File 'lib/faster_prime/sieve.rb', line 156

def mark_type_3(flags, f, g, d, l, b)
  x, y0 = f, g
  k0 = (3 * f * f - g * g - d) / 60 - l
  m = 1 << d
  while true
    while k0 >= b
      return if x <= y0
      k0 -= y0 + 15
      y0 += 30
    end
    k, y = k0, y0
    while k >= 0 && y < x
      flags[k] ^= m # mark
      k -= y + 15
      y += 30
    end
    k0 += x + 5
    x += 10
  end
end

.squarefree(flags, l, b) ⇒ Object 



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# File 'lib/faster_prime/sieve.rb', line 84

def squarefree(flags, l, b)
  SMALL_PRIMES.each do |q|
    q2 = q * q
    break if q2 > 60 * (l + b)
    next if q < 7
    minus_inv_60 = -Utils.mod_inv(60, q2)
    MODULO60.each do |d|
      # 60(l+x)+d == 0 (mod q2)
      # x = -d * 60^{-1} - l (mod q2)
      x = (d * minus_inv_60 - l) % q2
      m = ~(1 << d)
      while x < b
        flags[x] &= m
        x += q2
      end
    end
  end
end